When 2388, 4309 and 8151 are divided by a certain 3-digit number, the remainder in each case is the same. The remainder is:

Options

A. 39
B. 23
C. 19
D. 15

Answer

To find the common remainder when 2388, 4309, and 8151 are divided by a certain 3-digit number, we can follow these steps:

### Step 1: Calculate the differences between the numbers
First, we need to find the differences between the three numbers:

- Difference between 4309 and 2388:
\[ 4309 - 2388 = 1921 \]

- Difference between 8151 and 4309:
\[ 8151 - 4309 = 3842 \]

- Difference between 8151 and 2388:
\[ 8151 - 2388 = 5763 \]

### Step 2: Find the GCD of the differences
Next, we need to find the greatest common divisor (GCD) of these differences. We will use the GCD of 1921, 3842, and 5763.

1. **Finding GCD of 1921 and 3842:**
- 3842 = 2 × 1921
- Thus, GCD(1921, 3842) = 1921.

2. **Finding GCD of 1921 and 5763:**
- We can use the Euclidean algorithm:
\[ 5763 \div 1921 \approx 3 \quad \text{(3 times 1921 is 5763)} \]
- Remainder:
\[ 5763 - 3 \times 1921 = 0 \]
- Thus, GCD(1921, 5763) = 1921.

### Step 3: Factor the GCD
Now we will factor 1921 to find its divisors:
- 1921 can be factored as:
\[ 1921 = 17 \times 113 \]

### Step 4: Check for 3-digit divisors
The 3-digit factors of 1921 are:
- 113 (not a 3-digit number)
- 17 (not a 3-digit number)
- 1921 (not a 3-digit number)

Since we need a 3-digit divisor, we can check the divisors of 1921. The only 3-digit number that divides 1921 is 113.

### Step 5: Calculate the remainder
Now we need to find the common remainder when 2388, 4309, and 8151 are divided by 113.

1. **Finding remainder of 2388:**
\[ 2388 \div 113 \approx 21 \quad \text{(21 times 113 is 2373)} \]
- Remainder:
\[ 2388 - 2373 = 15 \]

2. **Finding remainder of 4309:**
\[ 4309 \div 113 \approx 38 \quad \text{(38 times 113 is 4294)} \]
- Remainder:
\[ 4309 - 4294 = 15 \]

3. **Finding remainder of 8151:**
\[ 8151 \div 113 \approx 72 \quad \text{(72 times 113 is 8156)} \]
- Remainder:
\[ 8151 - 8156 = -5 \quad \text{(which is equivalent to 15 when considering positive remainders)} \]

### Conclusion
The common remainder when 2388, 4309, and 8151 are divided by the 3-digit number is **15**.